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1 Purpose

D03PVF calculates a numerical flux function using Osher's Approximate Riemann Solver for the Euler equations in conservative form. It is designed primarily for use with the upwind discretization schemes D03PFF, D03PLF or D03PSF, but may also be applicable to other conservative upwind schemes requiring numerical flux functions.

2 Specification

3 Description

D03PVF calculates a numerical flux function at a single spatial point using Osher's Approximate Riemann Solver (see Hemker and Spekreijse (1986) and Pennington and Berzins (1994)) for the Euler equations (for a perfect gas) in conservative form. You must supply the left and right solution values at the point where the numerical flux is required, i.e., the initial left and right states of the Riemann problem defined below. In the routines D03PFF, D03PLF and D03PSF, the left and right solution values are derived automatically from the solution values at adjacent spatial points and supplied to the subroutine argument NUMFLX from which you may call D03PVF.

The Euler equations for a perfect gas in conservative form are:

∂U∂t+∂F∂x=0,

(1)

with

U=ρme and F=mm2ρ+γ-1e-m22ρmeρ+mργ-1e-m22ρ,

(2)

where ρ is the density, m is the momentum, e is the specific total energy, and γ is the (constant) ratio of specific heats. The pressure p is given by

p=γ-1e-ρu22,

(3)

where u=m/ρ is the velocity.

The routine calculates the Osher approximation to the numerical flux function FUL,UR=FU*UL,UR, where U=UL and U=UR are the left and right solution values, and U*UL,UR is the intermediate state ω0 arising from the similarity solution Uy,t=ωy/t of the Riemann problem defined by

∂U∂t+∂F∂y=0,

(4)

with U and F as in (2), and initial piecewise constant values U=UL for y<0 and U=UR for y>0. The spatial domain is -∞<y<∞, where y=0 is the point at which the numerical flux is required. Osher's solver carries out an integration along a path in the phase space of U consisting of subpaths which are piecewise parallel to the eigenvectors of the Jacobian of the PDE system. There are two variants of the Osher solver termed O (original) and P (physical), which differ in the order in which the subpaths are taken. The P-variant is generally more efficient, but in some rare cases may fail (see Hemker and Spekreijse (1986) for details). The parameter PATH specifies which variant is to be used. The algorithm for Osher's solver for the Euler equations is given in detail in the Appendix of Pennington and Berzins (1994).

5 Parameters

1: ULEFT(3) – REAL (KIND=nag_wp) arrayInput

On entry: ULEFTi must contain the left value of the component Ui, for i=1,2,3. That is, ULEFT1 must contain the left value of ρ, ULEFT2 must contain the left value of m and ULEFT3 must contain the left value of e.

On entry: URIGHTi must contain the right value of the component Ui, for i=1,2,3. That is, URIGHT1 must contain the right value of ρ, URIGHT2 must contain the right value of m and URIGHT3 must contain the right value of e.

On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.

On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

Note: if the left and/or right values of ρ or p (from (3)) are found to be negative, then the routine will terminate with an error exit (IFAIL=2). If the routine is being called from the NUMFLX etc., then a soft fail option (IFAIL=1 or -1) is recommended so that a recalculation of the current time step can be forced using the NUMFLX parameter IRES (see D03PFF or D03PLF).

6 Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

IFAIL=1

On entry,

GAMMA≤0.0,

or

PATH≠'O' or 'P'.

IFAIL=2

On entry,

the left and/or right density or pressure value is less than 0.0.

7 Accuracy

D03PVF performs an exact calculation of the Osher numerical flux function, and so the result will be accurate to machine precision.

8 Further Comments

D03PVF must only be used to calculate the numerical flux for the Euler equations in exactly the form given by (2), with ULEFTi and URIGHTi containing the left and right values of ρ,m and e, for i=1,2,3, respectively. It should be noted that Osher's scheme, in common with all Riemann solvers, may be unsuitable for some problems (see Quirk (1994) for examples). The time taken depends on the input parameter PATH and on the left and right solution values, since inclusion of each subpath depends on the signs of the eigenvalues. In general this cannot be determined in advance.